NAG Library Routine Document

, there is a danger of the result being totally inaccurate, as the error amplification factor grows in an essentially exponential manner; therefore the routine must fail.

4

References

NIST Digital Library of Mathematical Functions

5

Arguments

1: $x$ – Real (Kind=nag_wp)Input: On entry: the argument $x$ of the function.
2: $ifail$ – IntegerInput/Output: On entry: ifail must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$ . When the value $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $x = 〈value〉$ .
Constraint: $|x| \leq 〈value〉$ .
$|x|$ is too large for an accurate result to be returned and the function returns zero.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

Since the function is oscillatory, the absolute error rather than the relative error is important. Let

E

be the absolute error in the result and

δ

be the relative error in the argument. If

δ

is somewhat larger than the machine precision, then we have:

E ≃ |\frac{x}{\sqrt{2}} ({ber}_{1} x + {bei}_{1} x)| δ

(provided

E

is within machine bounds).

For small

x

the error amplification is insignificant and thus the absolute error is effectively bounded by the machine precision.

For medium and large

x

, the error behaviour is oscillatory and its amplitude grows like

\sqrt{\frac{x}{2 π}} e^{x / \sqrt{2}}

. Therefore it is not possible to calculate the function with any accuracy when

\sqrt{x} e^{x / \sqrt{2}} > \frac{\sqrt{2 π}}{δ}

. Note that this value of

x

is much smaller than the minimum value of

x

for which the function overflows.

8

Parallelism and Performance

s19aaf is not threaded in any implementation.

9

Further Comments

None.

10

Example

This example reads values of the argument

x

from a file, evaluates the function at each value of

x

and prints the results.

NAG Library Routine Document

s19aaf (kelvin_ber)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description